Journal of Nanomaterials

Volume 2015, Article ID 983413, 12 pages

http://dx.doi.org/10.1155/2015/983413

## Investigation of the Validity of the Universal Scaling Law on Linear Chains of Silver Nanoparticles

^{1}College of Art and Science, American University of Kuwait, 13034 Safat, Kuwait^{2}Department of Physics, Concordia University, Montréal, QC, Canada H4B 1R6

Received 14 December 2014; Accepted 4 February 2015

Academic Editor: Zhida Xu

Copyright © 2015 Mohammed Alsawafta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Due to the wide range of variation in the plasmonic characteristics of the metallic nanoparticles arranged in linear arrays, the optical spectra of these arrays provide a powerful platform for spectroscopic studies and biosensing applications. Due to the coupling effect between the interacting nanoparticles, the excited resonance mode is shifted with the interparticle separation. The change in the resonance energy of the coupled mode is expressed by the fractional plasmon shift which would normally follow a universal scaling behavior. Such a universal law has been successfully applied on a system of dimers under parallel polarization. It has been found that the plasmon shift decays exponentially over interparticle spacing. The decay length is independent of both the nanoparticle and dielectric properties of the surrounding medium. In this paper, the discrete dipole approximation (DDA) is used to examine the validity of extending the universal scaling law to linear chains of several interacting nanoparticles embedded in various host media for both parallel and perpendicular polarizations. Our calculations reveal that the decay length of both the coupled longitudinal mode (LM) and transverse modes (TM) is strongly dependent on the refractive index of the surrounding medium . The decay constant of the LM is linearly proportional to while the corresponding constant of the TM decays exponentially with . Upon changing the nanoparticle size, the change in the peak position of the LM decreases exponentially with the interparticle separation and hence, it obeys the universal law. The sensitivity of coupled LM to the nanoparticle size is more pronounced at both smaller nanoparticle sizes and separations. The sensitivity of the coupled TM to the nanoparticle size on the other hand changes linearly with the separation and therefore, the universal law does not apply in the case of the excited TM.

#### 1. Introduction

Transition metal nanoparticles have attracted considerable attention due to their unique electrical, optical, chemical, and magnetic properties as compared to their bulk counterparts [1, 2]. Noble-metal nanoparticles of a size smaller than the incident wavelength can effectively scatter and selectively absorb light at a certain wavelength in both visible and near-infrared regions [3, 4]. The fascinating optical properties of the metallic nanostructures originate from the excitation of the localized surface plasmon resonance (LSPR) [5, 6]. Due to the enhancement of the local field in the vicinity of the nanoparticles (hot spots), they are used in both biosensing [7, 8] and Surface Enhanced Raman Scattering (SERS) applications [9, 10].

Well-defined metallic nanoparticles arranged in several dimensional arrays provide an interesting opportunity to tune their optical properties over a wide range of optical parameters [11–13]. Periodic structures of nanoparticles can lead to a significant enhancement of the local electromagnetic field, which can be used to improve detection and characterization capabilities down to single-molecule level [13, 14]. Finite chains of metallic nanoparticles are considered as energy guides because of the possibility to transport their plasmonic energies along the chain axis [11, 13, 14]. The tuned plasmonic coupling between the interacting nanoparticles is an efficient way to distribute and direct the coupled energy through the plasmonic elements. In order to optimize the coupled plasmonic energy, one needs to understand the effect of various parameters on the collective plasmonic resonances. These parameters are the interparticle separation (), the number of the interacting nanoparticles (), the polarization states of the incident light, and the refractive index of the host media () where the nanoparticles are embedded. Depending on the spacing between the nanoparticles, two regimes of plasmonic coupling are considered: (i) when the nanoparticles are arranged in close-packed configurations, they interact via their near-fields and this leads to either a red- or blue-shift of the plasmonic band depending on the type of the incident light [11]. In general, this interaction results in enhanced LSPR and sensing capabilities of SERS. (ii) If the nanoparticles are further displaced by a distance comparable to the incident wavelength, they interact through their radiative fields [11, 15, 16]. These far-field couplings are important for the plasmon enhanced fluorescence of the adsorbed molecules on the nanoparticles surface [11].

The dependency of the plasmon coupling parameters on the interparticle spacing was experimentally studied for nanoparticles of different shapes arranged in various configurations. These nanoparticles were either a dimer of nanodiscs or single-sized spherical nanoparticles arranged in 1D array of different sizes [17]. These nanoparticles were embedded in host media of constant refractive index. No study is reported yet on the effect of the nanoparticle size and the dielectric properties of the host medium on the coupled plasmon resonance for linear chains composed of several plasmonic elements. It has been found that the coupled plasmon resonance wavelength of a dimer system shifts exponentially with the interparticle separation. This shift was expressed by the fractional plasmon shift leading to a universal scaling behaviour [17, 18]. The universality relies on the fact that the decay constant is independent of the nanoparticle size, shape, metal type, and host medium. Consequently, the proposed “plasmon ruler equation” [17, 18] was used to evaluate the spacing between a pair of nanoparticles from their measured plasmon shift. The universal scaling behaviour has been successfully applied to a trimer of nanospheres [19]. To the best of our knowledge, no study is reported on the validity of the scaling behavior for a linear chain of nanoparticles composed of several plasmonic elements.

The first purpose of this paper is to better understand and provide a full picture of the effect of several parameters on the near-field couplings. The parameters include the nanoparticle size, chain size and dielectric properties of the surrounding media at different polarization angles. The second purpose is to examine the validity of extending the universal scaling law to a linear chain that consists of several identical silver nanospheres. We believe that the results of the current study will provide a pathway to design nanoparticle chains for many technological applications. In order to perform the required calculations, the absorption spectra of linear silver chains consisting of spherical nanoparticles were simulated by using the discrete dipole approximation (DDA) method [20–22].

#### 2. The Discrete Dipole Approximation (DDA)

The DDA method is a numerical approximation used to solve Maxwell’s equations of the scattering problem of electromagnetic waves by metallic nanoparticles. Based on the induced electric dipole moment in the nanoparticles, DDA is used to calculate their optical cross-sections of different shapes and sizes in complex surrounding media. It involves replacing each nanoparticle by a three-dimensional array of polarizable points arranged in a cubic configuration whose side length is equal to the interdipole spacing. The optical properties of the nanoparticles are determined by three factors: (1) the incident wavelength, (2) the polarizability of the nanoparticles, and (3) the mutual interaction between both dipoles within the same nanoparticle and other dipoles in the nearby nanoparticles. The mathematical formulation of DDA is beyond the scope of the current study and is fully described elsewhere [20–22]. DDA is tolerant regarding the target geometry and size. The only limitation to be considered is that the interdipole separation should be smaller than the incident wavelength and any other structural parameters. The accuracy of DDA is widely accepted when a large number of dipoles are used to properly mimic the geometrical parameters of the target. In this study, the desired output of DDA is the absorption cross-section of the nanoparticle, normalized to its geometrical cross-section, which yields the corresponding efficiency ().

#### 3. Target Geometry and Orientation

The target under investigation is represented by 1D array of silver nanoparticles of similar polarizabilities. The selected nanoparticles are spherical in shape, different in size, placed at various separations (), and embedded in different host media. is the border-to-border distance between the nearest-neighbor nanoparticles and it is usually defined in terms of the sphere radius. The optical spectra of many-body interaction problems are calculated by using the DDA method.

The nanoparticle chains are irradiated with an electromagnetic plane wave. The structure and the orientation of the chains relative to the incident field are shown in Figure 1. The plane of incidence is set to be the - plane and the incident electric field () is aligned along the -axis. The linearly polarized (-polarized) incident light determines the polarization angle () between and the chain axis. At oblique angles, the incident electric field has two components: one parallel to the chain axis and the other one oriented along the - plane. In this case, the observation of dipolar plasmonic modes of different origins is possible under distinct combinations of many parameters. On the other hand, the electric field of the -polarized light has one component perpendicular to the chain axis at any angle of incidence, and hence, only one plasmonic band of transversal character is expected. In the case of unpolarized light, the spectra are calculated as an average over the two polarizations directions, and the spectrum exhibits all LSPR modes.